Does anyone know if there is a classification of the subgroups of the real numbers taken under addition? If not can anyone point me in the directiong of any papers/materials which discuss properties of or interesting facts about these subgroups?

@Hans, for (1) use Steinhaus' interior point theorem (immediate); for (2) consider the "distrubution function" $F(x)=m^*([0,x]\cap G)$; for (3) construct a nonmeasurable additive functional whose kernel gives such a group. I think (1) and (3) should be available in some articles.
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Syang ChenFeb 12 '12 at 2:46

Every torsion-free abelian group of cardinality at most $2^\omega$ is isomorphic to a subgroup of the reals. (To see this, note that any such group can be embedded in a divisible torsion-free group of the same cardinality, i.e., a vector space over $\mathbb Q$, which can in turn be embedded in any other vector space over $\mathbb Q$ of the same or greater dimension.) Since already the structure of rank 2 abelian groups is hopelessly complicated, you are not going to find any sensible classification.

I'm surprised no one has yet stated the most obvious fact (though I guess Xianghong's answer comes pretty close), which is that an additive subgroup of the reals is either of the form $a\mathbb{Z}$ or is dense in the real line (an obvious consequence from division with remainder).

And although this is in another direction of hopeless complication, adding a finite rank additive subgroup to the reals with the field structure allows one to define the integers. To see this, take the set of reals for which multiplication by $r$ is a map from $G \rightarrow G.$ Take the fraction field of this set. This is a finite degree extension of $\mathbb{Q}.$ Now results of J. Robinson give the definability of the integers. So, this is another instance of hopeless complication.
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James FreitagMar 29 '11 at 17:20

If you're interested in topological classification, then this might be useful: Farah and Solecki - Borel subgroups of Polish groups, Advances in Mathematics 199, 2006, 499-541.

Among a lot of other things, one of their results shows that for any countable ordinals $\alpha \neq 2$ and $\beta \geq 2$, there are $\Pi_{\alpha}^{0}$-complete and $\Sigma_{\beta}^{0}$-complete additive subgroups of any uncountable polish group. For connected abelian polish groups, this was previously shown by Mauldin by refining a result of Klee.